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				<span style="position: absolute;left:15px;bottom:15px;width:90%;"><font class="view-text" style="color:#fcfcfc;font-size:25px">题解 P3216 【[HNOI2011]数学作业】</font><br><a href="/tags/2020/" class="tag"><span  style="background-color: rgb(52, 152, 219);">2020</span></a>&nbsp;<a href="/tags/矩阵快速幂/" class="tag"><span  style="background-color: rgb(231, 76, 60);">矩阵快速幂</span></a>&nbsp;<a href="/tags/题解/" class="tag"><span  style="background-color: rgb(82, 196, 26);">题解</span></a></span>
			</div>
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                <h2 id="_1">题意</h2>
<p>
<script type="math/tex">\text{Concatenate}(n)</script> 是将 <script type="math/tex">1 \sim n</script> 所有正整数 顺序连接起来得到的数，计算 <script type="math/tex">\text{Concatenate}(n) \bmod m</script> 的值。</p>
<!--more-->
<h2 id="_2">题解</h2>
<p>对与位数相同的数，用<script type="math/tex">k</script>表示位数，用<script type="math/tex">f[x]</script>表示从<script type="math/tex">10^k</script>一直写到<script type="math/tex">x</script>的值，则有:
<script type="math/tex; mode=display">
f[x]=f[x-1]\times 10^{k+1}+x\pmod m
</script>
就相当于在原串上拼接了<script type="math/tex">x</script>，但看到<script type="math/tex">1\le n \le 10^{18}</script>的数据范围<sub>~当场就放弃了</sub>~，很明显我们必须优化上面的柿子，不然程序能T飞到天上去</p>
<p>这里我们需要引入矩阵乘法的概念</p>
<blockquote>
<p>矩阵相乘最重要的方法是一般矩阵乘积。它只有在第一个矩阵的列数（column）和第二个矩阵的行数（row）相同时才有意义 [1]  。一般单指矩阵乘积时，指的便是一般矩阵乘积。一个<script type="math/tex">m×n</script>的矩阵就是<script type="math/tex">m×n</script>个数排成<script type="math/tex">m</script>行<script type="math/tex">n</script>列的一个数阵。由于它把许多数据紧凑地集中到了一起，所以有时候可以简便地表示一些复杂的模型，如电力系统网络模型。 [2] --百度百科</p>
<p>设<script type="math/tex">A</script>为<script type="math/tex">m\times p</script> 的矩阵，<script type="math/tex">B</script>为<script type="math/tex">p\times n</script> 的矩阵，那么称<script type="math/tex">m\times n</script>的矩阵<script type="math/tex">C</script>为矩阵<script type="math/tex">A</script>与<script type="math/tex">B</script>的乘积，记作 <script type="math/tex">C=AB</script>，其中矩阵C中的第 <script type="math/tex">i</script>行第<script type="math/tex">j</script> 列元素可以表示为：<script type="math/tex">(AB)_{ij}=\sum_{k=1}^pa_{ik}b_{kj}</script>
</p>
</blockquote>
<p>
<script type="math/tex; mode=display">\begin{matrix}
 f[x+1]&=&f[x]\times 10^{k+1}  &+x&+1 \\ 
 x+1&=&0  &+x&+1 \\ 
 1&=&0  &+0&+1 \\ 
\end{matrix}
</script>
那么矩阵也很容易构造了
<script type="math/tex; mode=display">
\text{设}P=\begin{bmatrix}
10^{k+1}&1&1\\
0 &1&1\\
0&0&1
\end{bmatrix}
</script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{pmatrix}
g[x]\\ x\\ 1
\end{pmatrix}
=P\times 
\begin{pmatrix}
g[x-1]\\ x-1\\ 1
\end{pmatrix}
=P \times P \times 
\begin{pmatrix}
g[x-2]\\ 
x-2\\  
1
\end{pmatrix}
=
\dots=
P^{x-10^k}\times 
\begin{pmatrix}
10^k\\ 10^k\\ 1
\end{pmatrix}
</script>
</p>
<p>对于由于矩阵也有结合律，式中用快速幂处理<script type="math/tex">P</script>的次方就行了</p>
<h2 id="_3">代码</h2>
<p><div class="highlight"><pre><span></span><code><span class="linenos" data-linenos="  1 "></span> <span class="c1">//#pragma GCC optimize(3,&quot;Ofast&quot;,&quot;inline&quot;)</span>
<span class="linenos" data-linenos="  2 "></span><span class="cp">#include</span><span class="cpf">&lt;bits/stdc++.h&gt;</span><span class="cp"></span>
<span class="linenos" data-linenos="  3 "></span><span class="k">namespace</span> <span class="n">in</span><span class="p">{</span>
<span class="linenos" data-linenos="  4 "></span>    <span class="kt">char</span> <span class="n">buf</span><span class="p">[</span><span class="mi">1</span><span class="o">&lt;&lt;</span><span class="mi">21</span><span class="p">],</span><span class="o">*</span><span class="n">p1</span><span class="o">=</span><span class="n">buf</span><span class="p">,</span><span class="o">*</span><span class="n">p2</span><span class="o">=</span><span class="n">buf</span><span class="p">;</span>
<span class="linenos" data-linenos="  5 "></span>    <span class="kr">inline</span> <span class="kt">int</span> <span class="n">getc</span><span class="p">(){</span>
<span class="linenos" data-linenos="  6 "></span>        <span class="k">return</span> <span class="n">p1</span><span class="o">==</span><span class="n">p2</span><span class="o">&amp;&amp;</span><span class="p">(</span><span class="n">p2</span><span class="o">=</span><span class="p">(</span><span class="n">p1</span><span class="o">=</span><span class="n">buf</span><span class="p">)</span><span class="o">+</span><span class="n">fread</span><span class="p">(</span><span class="n">buf</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="o">&lt;&lt;</span><span class="mi">21</span><span class="p">,</span><span class="n">stdin</span><span class="p">),</span><span class="n">p1</span><span class="o">==</span><span class="n">p2</span><span class="p">)</span><span class="o">?</span><span class="nl">EOF</span><span class="p">:</span><span class="o">*</span><span class="n">p1</span><span class="o">++</span><span class="p">;</span>
<span class="linenos" data-linenos="  7 "></span>    <span class="p">}</span>
<span class="linenos" data-linenos="  8 "></span>    <span class="k">template</span> <span class="o">&lt;</span><span class="k">typename</span> <span class="nc">T</span><span class="o">&gt;</span><span class="kr">inline</span> <span class="kt">void</span> <span class="n">read</span><span class="p">(</span><span class="n">T</span><span class="o">&amp;</span> <span class="n">t</span><span class="p">){</span>
<span class="linenos" data-linenos="  9 "></span>        <span class="n">t</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="kt">int</span> <span class="n">f</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="kt">char</span> <span class="n">ch</span><span class="o">=</span><span class="n">getc</span><span class="p">();</span>
<span class="linenos" data-linenos=" 10 "></span>        <span class="k">while</span> <span class="p">(</span><span class="o">!</span><span class="n">isdigit</span><span class="p">(</span><span class="n">ch</span><span class="p">)){</span>
<span class="linenos" data-linenos=" 11 "></span>            <span class="k">if</span><span class="p">(</span><span class="n">ch</span><span class="o">==</span><span class="sc">&#39;-&#39;</span><span class="p">)</span><span class="n">f</span> <span class="o">=</span> <span class="mi">1</span><span class="p">;</span>
<span class="linenos" data-linenos=" 12 "></span>            <span class="n">ch</span><span class="o">=</span><span class="n">getc</span><span class="p">();</span>
<span class="linenos" data-linenos=" 13 "></span>        <span class="p">}</span>
<span class="linenos" data-linenos=" 14 "></span>        <span class="k">while</span><span class="p">(</span><span class="n">isdigit</span><span class="p">(</span><span class="n">ch</span><span class="p">)){</span>
<span class="linenos" data-linenos=" 15 "></span>            <span class="n">t</span><span class="o">=</span><span class="n">t</span><span class="o">*</span><span class="mi">10</span><span class="o">+</span><span class="n">ch</span><span class="mi">-48</span><span class="p">;</span>
<span class="linenos" data-linenos=" 16 "></span>            <span class="n">ch</span> <span class="o">=</span> <span class="n">getc</span><span class="p">();</span>
<span class="linenos" data-linenos=" 17 "></span>        <span class="p">}</span>
<span class="linenos" data-linenos=" 18 "></span>        <span class="k">if</span><span class="p">(</span><span class="n">f</span><span class="p">)</span><span class="n">t</span><span class="o">=-</span><span class="n">t</span><span class="p">;</span>
<span class="linenos" data-linenos=" 19 "></span>    <span class="p">}</span>
<span class="linenos" data-linenos=" 20 "></span>    <span class="k">template</span> <span class="o">&lt;</span><span class="k">typename</span> <span class="nc">T</span><span class="p">,</span><span class="k">typename</span><span class="p">...</span> <span class="n">Args</span><span class="o">&gt;</span> <span class="kr">inline</span> <span class="kt">void</span> <span class="n">read</span><span class="p">(</span><span class="n">T</span><span class="o">&amp;</span> <span class="n">t</span><span class="p">,</span> <span class="n">Args</span><span class="o">&amp;</span><span class="p">...</span> <span class="n">args</span><span class="p">){</span>
<span class="linenos" data-linenos=" 21 "></span>        <span class="n">read</span><span class="p">(</span><span class="n">t</span><span class="p">);</span><span class="n">read</span><span class="p">(</span><span class="n">args</span><span class="p">...);</span>
<span class="linenos" data-linenos=" 22 "></span>    <span class="p">}</span>
<span class="linenos" data-linenos=" 23 "></span><span class="p">}</span>
<span class="linenos" data-linenos=" 24 "></span><span class="k">namespace</span> <span class="n">out</span><span class="p">{</span>
<span class="linenos" data-linenos=" 25 "></span>    <span class="kt">char</span> <span class="n">buffer</span><span class="p">[</span><span class="mi">1</span><span class="o">&lt;&lt;</span><span class="mi">21</span><span class="p">];</span>
<span class="linenos" data-linenos=" 26 "></span>    <span class="kt">int</span> <span class="n">p1</span><span class="o">=</span><span class="mi">-1</span><span class="p">;</span>
<span class="linenos" data-linenos=" 27 "></span>    <span class="k">const</span> <span class="kt">int</span> <span class="n">p2</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="o">&lt;&lt;</span><span class="mi">21</span><span class="p">)</span><span class="mi">-1</span><span class="p">;</span>
<span class="linenos" data-linenos=" 28 "></span>    <span class="kr">inline</span> <span class="kt">void</span> <span class="n">flush</span><span class="p">()</span> <span class="p">{</span>
<span class="linenos" data-linenos=" 29 "></span>        <span class="n">fwrite</span><span class="p">(</span><span class="n">buffer</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">p1</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">stdout</span><span class="p">),</span>
<span class="linenos" data-linenos=" 30 "></span>        <span class="n">p1</span><span class="o">=</span><span class="mi">-1</span><span class="p">;</span>
<span class="linenos" data-linenos=" 31 "></span>    <span class="p">}</span>
<span class="linenos" data-linenos=" 32 "></span>    <span class="kr">inline</span> <span class="kt">void</span> <span class="n">putc</span><span class="p">(</span><span class="k">const</span> <span class="kt">char</span> <span class="o">&amp;</span><span class="n">x</span><span class="p">)</span> <span class="p">{</span>
<span class="linenos" data-linenos=" 33 "></span>        <span class="k">if</span><span class="p">(</span><span class="n">p1</span><span class="o">==</span><span class="n">p2</span><span class="p">)</span><span class="n">flush</span><span class="p">();</span>
<span class="linenos" data-linenos=" 34 "></span>        <span class="n">buffer</span><span class="p">[</span><span class="o">++</span><span class="n">p1</span><span class="p">]</span><span class="o">=</span><span class="n">x</span><span class="p">;</span>
<span class="linenos" data-linenos=" 35 "></span>    <span class="p">}</span>
<span class="linenos" data-linenos=" 36 "></span>    <span class="k">template</span> <span class="o">&lt;</span><span class="k">typename</span> <span class="nc">T</span><span class="o">&gt;</span><span class="kt">void</span> <span class="n">write</span><span class="p">(</span><span class="n">T</span> <span class="n">x</span><span class="p">)</span> <span class="p">{</span>
<span class="linenos" data-linenos=" 37 "></span>        <span class="k">static</span> <span class="kt">char</span> <span class="n">buf</span><span class="p">[</span><span class="mi">15</span><span class="p">];</span>
<span class="linenos" data-linenos=" 38 "></span>        <span class="k">static</span> <span class="kt">int</span> <span class="n">len</span><span class="o">=</span><span class="mi">-1</span><span class="p">;</span>
<span class="linenos" data-linenos=" 39 "></span>        <span class="k">if</span><span class="p">(</span><span class="n">x</span><span class="o">&gt;=</span><span class="mi">0</span><span class="p">){</span>
<span class="linenos" data-linenos=" 40 "></span>            <span class="k">do</span><span class="p">{</span>
<span class="linenos" data-linenos=" 41 "></span>                <span class="n">buf</span><span class="p">[</span><span class="o">++</span><span class="n">len</span><span class="p">]</span><span class="o">=</span><span class="n">x</span><span class="o">%</span><span class="mi">10</span><span class="o">+</span><span class="mi">48</span><span class="p">,</span><span class="n">x</span><span class="o">/=</span><span class="mi">10</span><span class="p">;</span>
<span class="linenos" data-linenos=" 42 "></span>            <span class="p">}</span><span class="k">while</span> <span class="p">(</span><span class="n">x</span><span class="p">);</span>
<span class="linenos" data-linenos=" 43 "></span>        <span class="p">}</span><span class="k">else</span><span class="p">{</span>
<span class="linenos" data-linenos=" 44 "></span>            <span class="n">putc</span><span class="p">(</span><span class="sc">&#39;-&#39;</span><span class="p">);</span>
<span class="linenos" data-linenos=" 45 "></span>            <span class="k">do</span> <span class="p">{</span>
<span class="linenos" data-linenos=" 46 "></span>                <span class="n">buf</span><span class="p">[</span><span class="o">++</span><span class="n">len</span><span class="p">]</span><span class="o">=-</span><span class="p">(</span><span class="n">x</span><span class="o">%</span><span class="mi">10</span><span class="p">)</span><span class="o">+</span><span class="mi">48</span><span class="p">,</span><span class="n">x</span><span class="o">/=</span><span class="mi">10</span><span class="p">;</span>
<span class="linenos" data-linenos=" 47 "></span>            <span class="p">}</span><span class="k">while</span><span class="p">(</span><span class="n">x</span><span class="p">);</span>
<span class="linenos" data-linenos=" 48 "></span>        <span class="p">}</span>
<span class="linenos" data-linenos=" 49 "></span>        <span class="k">while</span> <span class="p">(</span><span class="n">len</span><span class="o">&gt;=</span><span class="mi">0</span><span class="p">)</span>
<span class="linenos" data-linenos=" 50 "></span>            <span class="n">putc</span><span class="p">(</span><span class="n">buf</span><span class="p">[</span><span class="n">len</span><span class="p">]),</span><span class="o">--</span><span class="n">len</span><span class="p">;</span>
<span class="linenos" data-linenos=" 51 "></span>    <span class="p">}</span>
<span class="linenos" data-linenos=" 52 "></span><span class="p">}</span>
<span class="linenos" data-linenos=" 53 "></span><span class="c1">//fread快读快输</span>
<span class="linenos" data-linenos=" 54 "></span><span class="c1">//让你的代码看上去更高级（逃 </span>
<span class="linenos" data-linenos=" 55 "></span><span class="cp">#define int unsigned long long</span>
<span class="linenos" data-linenos=" 56 "></span><span class="k">using</span> <span class="k">namespace</span> <span class="n">std</span><span class="p">;</span>
<span class="linenos" data-linenos=" 57 "></span><span class="kt">int</span> <span class="n">mod</span><span class="p">;</span>
<span class="linenos" data-linenos=" 58 "></span><span class="k">struct</span> <span class="nc">mat</span><span class="p">{</span><span class="c1">//矩阵</span>
<span class="linenos" data-linenos=" 59 "></span>    <span class="kt">int</span> <span class="n">m</span><span class="p">,</span><span class="n">n</span><span class="p">;</span><span class="c1">//矩阵的行数、列数</span>
<span class="linenos" data-linenos=" 60 "></span>    <span class="kt">int</span> <span class="n">a</span><span class="p">[</span><span class="mi">10</span><span class="p">][</span><span class="mi">10</span><span class="p">];</span>
<span class="linenos" data-linenos=" 61 "></span>    <span class="kt">int</span> <span class="o">*</span><span class="k">operator</span><span class="p">[](</span><span class="k">const</span> <span class="kt">int</span> <span class="n">i</span><span class="p">){</span>
<span class="linenos" data-linenos=" 62 "></span>        <span class="k">return</span> <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
<span class="linenos" data-linenos=" 63 "></span>    <span class="p">}</span>
<span class="linenos" data-linenos=" 64 "></span>    <span class="n">mat</span><span class="p">(){</span>
<span class="linenos" data-linenos=" 65 "></span>        <span class="n">memset</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="k">sizeof</span> <span class="n">a</span><span class="p">);</span>
<span class="linenos" data-linenos=" 66 "></span>        <span class="n">m</span><span class="o">=</span><span class="n">n</span><span class="o">=</span><span class="mi">3</span><span class="p">;</span>
<span class="linenos" data-linenos=" 67 "></span>    <span class="p">}</span>
<span class="linenos" data-linenos=" 68 "></span>    <span class="n">mat</span><span class="p">(</span><span class="kt">int</span> <span class="n">hang</span><span class="p">,</span><span class="kt">int</span> <span class="n">lie</span><span class="p">,</span><span class="kt">int</span> <span class="o">*</span><span class="n">num</span><span class="p">){</span>
<span class="linenos" data-linenos=" 69 "></span>        <span class="n">m</span><span class="o">=</span><span class="n">hang</span><span class="p">,</span><span class="n">n</span><span class="o">=</span><span class="n">lie</span><span class="p">;</span>
<span class="linenos" data-linenos=" 70 "></span>        <span class="k">for</span><span class="p">(</span><span class="kt">int</span> <span class="n">i</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">tot</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="n">i</span><span class="o">&lt;=</span><span class="n">hang</span><span class="p">;</span><span class="n">i</span><span class="o">++</span><span class="p">)</span>
<span class="linenos" data-linenos=" 71 "></span>            <span class="k">for</span><span class="p">(</span><span class="kt">int</span> <span class="n">j</span><span class="o">=</span><span class="mi">1</span><span class="p">;</span><span class="n">j</span><span class="o">&lt;=</span><span class="n">lie</span><span class="p">;</span><span class="n">j</span><span class="o">++</span><span class="p">)</span>
<span class="linenos" data-linenos=" 72 "></span>                <span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="o">=</span><span class="n">num</span><span class="p">[</span><span class="n">tot</span><span class="o">++</span><span class="p">];</span>
<span class="linenos" data-linenos=" 73 "></span>    <span class="p">}</span> 
<span class="linenos" data-linenos=" 74 "></span>    <span class="n">mat</span><span class="p">(</span><span class="kt">int</span> <span class="n">hang</span><span class="p">,</span><span class="kt">int</span> <span class="n">lie</span><span class="p">){</span>
<span class="linenos" data-linenos=" 75 "></span>        <span class="n">m</span><span class="o">=</span><span class="n">hang</span><span class="p">,</span><span class="n">n</span><span class="o">=</span><span class="n">lie</span><span class="p">;</span>
<span class="linenos" data-linenos=" 76 "></span>        <span class="n">memset</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="k">sizeof</span> <span class="n">a</span><span class="p">);</span>
<span class="linenos" data-linenos=" 77 "></span>    <span class="p">}</span> 
<span class="linenos" data-linenos=" 78 "></span>    <span class="n">mat</span> <span class="k">operator</span><span class="o">*</span><span class="p">(</span><span class="n">mat</span> <span class="n">b</span><span class="p">)</span><span class="k">const</span><span class="p">{</span><span class="c1">//重载乘号 </span>
<span class="linenos" data-linenos=" 79 "></span>        <span class="c1">//保证n=b.m</span>
<span class="linenos" data-linenos=" 80 "></span>        <span class="n">mat</span> <span class="nf">c</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="n">b</span><span class="p">.</span><span class="n">n</span><span class="p">);</span>
<span class="linenos" data-linenos=" 81 "></span>        <span class="k">for</span><span class="p">(</span><span class="kt">int</span> <span class="n">i</span><span class="o">=</span><span class="mi">1</span><span class="p">;</span><span class="n">i</span><span class="o">&lt;=</span><span class="n">n</span><span class="p">;</span><span class="n">i</span><span class="o">++</span><span class="p">)</span>
<span class="linenos" data-linenos=" 82 "></span>        <span class="k">for</span><span class="p">(</span><span class="kt">int</span> <span class="n">j</span><span class="o">=</span><span class="mi">1</span><span class="p">;</span><span class="n">j</span><span class="o">&lt;=</span><span class="n">b</span><span class="p">.</span><span class="n">n</span><span class="p">;</span><span class="n">j</span><span class="o">++</span><span class="p">)</span>
<span class="linenos" data-linenos=" 83 "></span>        <span class="k">for</span><span class="p">(</span><span class="kt">int</span> <span class="n">k</span><span class="o">=</span><span class="mi">1</span><span class="p">;</span><span class="n">k</span><span class="o">&lt;=</span><span class="n">m</span><span class="p">;</span><span class="n">k</span><span class="o">++</span><span class="p">)</span> 
<span class="linenos" data-linenos=" 84 "></span>            <span class="n">c</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="o">+=</span><span class="n">a</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">b</span><span class="p">[</span><span class="n">k</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="o">%</span><span class="n">mod</span><span class="p">,</span><span class="n">c</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="o">%=</span><span class="n">mod</span><span class="p">;</span>
<span class="linenos" data-linenos=" 85 "></span>        <span class="k">return</span> <span class="n">c</span><span class="p">;</span>
<span class="linenos" data-linenos=" 86 "></span>    <span class="p">}</span>
<span class="linenos" data-linenos=" 87 "></span><span class="p">};</span>
<span class="linenos" data-linenos=" 88 "></span><span class="kt">int</span> <span class="n">n</span><span class="p">;</span>
<span class="linenos" data-linenos=" 89 "></span><span class="n">mat</span> <span class="nf">fpow</span><span class="p">(</span><span class="n">mat</span> <span class="n">a</span><span class="p">,</span><span class="kt">int</span> <span class="n">b</span><span class="p">){</span><span class="c1">//矩阵快速幂 </span>
<span class="linenos" data-linenos=" 90 "></span>    <span class="kt">int</span> <span class="n">tmp</span><span class="p">[]</span><span class="o">=</span><span class="p">{</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">};</span><span class="c1">//单位矩阵 </span>
<span class="linenos" data-linenos=" 91 "></span>    <span class="n">mat</span> <span class="n">ans</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="n">tmp</span><span class="p">);</span>
<span class="linenos" data-linenos=" 92 "></span>    <span class="k">while</span><span class="p">(</span><span class="n">b</span><span class="p">){</span>
<span class="linenos" data-linenos=" 93 "></span>        <span class="k">if</span><span class="p">(</span><span class="n">b</span><span class="o">&amp;</span><span class="mi">1</span><span class="p">)</span><span class="n">ans</span><span class="o">=</span><span class="n">ans</span><span class="o">*</span><span class="n">a</span><span class="p">;</span>
<span class="linenos" data-linenos=" 94 "></span>        <span class="n">a</span><span class="o">=</span><span class="n">a</span><span class="o">*</span><span class="n">a</span><span class="p">;</span><span class="n">b</span><span class="o">&gt;&gt;=</span><span class="mi">1</span><span class="p">;</span>
<span class="linenos" data-linenos=" 95 "></span>    <span class="p">}</span><span class="k">return</span> <span class="n">ans</span><span class="p">;</span>
<span class="linenos" data-linenos=" 96 "></span><span class="p">}</span>
<span class="linenos" data-linenos=" 97 "></span><span class="kt">int</span> <span class="nf">fpow</span><span class="p">(</span><span class="kt">int</span> <span class="n">a</span><span class="p">,</span><span class="kt">int</span> <span class="n">b</span><span class="p">){</span><span class="c1">//整数快速幂 </span>
<span class="linenos" data-linenos=" 98 "></span>    <span class="kt">int</span> <span class="n">ans</span><span class="o">=</span><span class="mi">1</span><span class="p">;</span>
<span class="linenos" data-linenos=" 99 "></span>    <span class="k">while</span><span class="p">(</span><span class="n">b</span><span class="p">){</span>
<span class="linenos" data-linenos="100 "></span>        <span class="k">if</span><span class="p">(</span><span class="n">b</span><span class="o">&amp;</span><span class="mi">1</span><span class="p">)</span><span class="n">ans</span><span class="o">=</span><span class="n">ans</span><span class="o">*</span><span class="n">a</span><span class="o">%</span><span class="n">mod</span><span class="p">;</span>
<span class="linenos" data-linenos="101 "></span>        <span class="n">a</span><span class="o">=</span><span class="n">a</span><span class="o">*</span><span class="n">a</span><span class="o">%</span><span class="n">mod</span><span class="p">;</span><span class="n">b</span><span class="o">&gt;&gt;=</span><span class="mi">1</span><span class="p">;</span>
<span class="linenos" data-linenos="102 "></span>    <span class="p">}</span><span class="k">return</span> <span class="n">ans</span><span class="p">;</span>
<span class="linenos" data-linenos="103 "></span><span class="p">}</span>
<span class="linenos" data-linenos="104 "></span><span class="kt">int</span> <span class="n">ans</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span><span class="n">already</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span>
<span class="linenos" data-linenos="105 "></span><span class="c1">//   答案 已有位数 </span>
<span class="linenos" data-linenos="106 "></span><span class="cp">#define pow lpow</span>
<span class="linenos" data-linenos="107 "></span><span class="kt">int</span> <span class="nf">lpow</span><span class="p">(</span><span class="kt">int</span> <span class="n">a</span><span class="p">,</span><span class="kt">int</span> <span class="n">b</span><span class="p">){</span>
<span class="linenos" data-linenos="108 "></span>    <span class="kt">int</span> <span class="n">ans</span><span class="o">=</span><span class="mi">1</span><span class="p">;</span>
<span class="linenos" data-linenos="109 "></span>    <span class="k">while</span><span class="p">(</span><span class="n">b</span><span class="p">){</span>
<span class="linenos" data-linenos="110 "></span>        <span class="k">if</span><span class="p">(</span><span class="n">b</span><span class="o">&amp;</span><span class="mi">1</span><span class="p">)</span><span class="n">ans</span><span class="o">=</span><span class="n">ans</span><span class="o">*</span><span class="n">a</span><span class="p">;</span>
<span class="linenos" data-linenos="111 "></span>        <span class="n">a</span><span class="o">=</span><span class="n">a</span><span class="o">*</span><span class="n">a</span><span class="p">;</span><span class="n">b</span><span class="o">&gt;&gt;=</span><span class="mi">1</span><span class="p">;</span>
<span class="linenos" data-linenos="112 "></span>    <span class="p">}</span><span class="k">return</span> <span class="n">ans</span><span class="p">;</span>
<span class="linenos" data-linenos="113 "></span>
<span class="linenos" data-linenos="114 "></span><span class="p">}</span>
<span class="linenos" data-linenos="115 "></span><span class="kt">signed</span> <span class="nf">main</span><span class="p">(){</span>
<span class="linenos" data-linenos="116 "></span>    <span class="n">in</span><span class="o">::</span><span class="n">read</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">mod</span><span class="p">);</span>
<span class="linenos" data-linenos="117 "></span>    <span class="k">for</span><span class="p">(</span><span class="kt">int</span> <span class="n">k</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="n">pow</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="n">k</span><span class="p">)</span><span class="o">&lt;=</span><span class="n">n</span><span class="p">;</span><span class="n">k</span><span class="o">++</span><span class="p">){</span><span class="c1">//枚举K </span>
<span class="linenos" data-linenos="118 "></span>        <span class="c1">//printf(&quot;%d\n&quot;,k);</span>
<span class="linenos" data-linenos="119 "></span>        <span class="kt">int</span> <span class="n">t</span><span class="p">[]</span><span class="o">=</span><span class="p">{</span><span class="n">fpow</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="n">k</span><span class="p">),</span><span class="n">fpow</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="n">k</span><span class="p">),</span><span class="mi">1</span><span class="p">};</span>
<span class="linenos" data-linenos="120 "></span>        <span class="n">mat</span> <span class="n">tmp</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">t</span><span class="p">);</span>
<span class="linenos" data-linenos="121 "></span>        <span class="kt">int</span> <span class="n">tt</span><span class="p">[]</span><span class="o">=</span><span class="p">{</span><span class="n">fpow</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">};</span>
<span class="linenos" data-linenos="122 "></span>        <span class="n">mat</span> <span class="n">p</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="n">tt</span><span class="p">);</span>
<span class="linenos" data-linenos="123 "></span>        <span class="c1">//printf(&quot; %lld\n&quot;,(min(pow(10,k+1),n+1)-pow(10,k))*(k+1));</span>
<span class="linenos" data-linenos="124 "></span>        <span class="n">ans</span><span class="o">*=</span><span class="n">fpow</span><span class="p">(</span><span class="mi">10</span><span class="p">,(</span><span class="n">min</span><span class="p">((</span><span class="kt">int</span><span class="p">)</span><span class="n">pow</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">-</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="n">pow</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="n">k</span><span class="p">))</span><span class="o">*</span><span class="p">(</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">));</span>
<span class="linenos" data-linenos="125 "></span>        <span class="n">ans</span><span class="o">%=</span><span class="n">mod</span><span class="p">;</span>
<span class="linenos" data-linenos="126 "></span>        <span class="c1">//printf(&quot; %lld\n&quot;,min((int)pow(10,k+1)-1,n)-(int)pow(10,k));</span>
<span class="linenos" data-linenos="127 "></span>        <span class="n">ans</span><span class="o">+=</span><span class="p">((</span><span class="n">fpow</span><span class="p">(</span><span class="n">p</span><span class="p">,(</span><span class="kt">int</span><span class="p">)</span><span class="n">min</span><span class="p">((</span><span class="kt">int</span><span class="p">)</span><span class="n">pow</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="n">k</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="mi">-1</span><span class="p">,</span><span class="n">n</span><span class="p">)</span><span class="o">-</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="n">pow</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="n">k</span><span class="p">))</span><span class="o">*</span><span class="n">tmp</span><span class="p">))[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">];</span>
<span class="linenos" data-linenos="128 "></span>        <span class="n">ans</span><span class="o">%=</span><span class="n">mod</span><span class="p">;</span>
<span class="linenos" data-linenos="129 "></span>    <span class="p">}</span>
<span class="linenos" data-linenos="130 "></span>    <span class="n">out</span><span class="o">::</span><span class="n">write</span><span class="p">(</span><span class="n">ans</span><span class="p">);</span>
<span class="linenos" data-linenos="131 "></span>    <span class="n">out</span><span class="o">::</span><span class="n">flush</span><span class="p">();</span>
<span class="linenos" data-linenos="132 "></span>    <span class="k">return</span> <span class="mi">0</span><span class="p">;</span>
<span class="linenos" data-linenos="133 "></span><span class="p">}</span>
</code></pre></div>
其实这题思维难度并不大，只是代码细节<strong>有点</strong>多</p>
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    <span><font style="font-weight: bold">题解 P3216 【[HNOI2011]数学作业】</font></span>
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